Math, asked by jimzkijurj, 1 year ago

Use the Division Algorithm to establish the following:
(a) The square of any integer is either of the form 3k or 3k + 1.
(b) The cube of any integer has one of the forms: 9k, 9k + 1, or 9k + 8.

Answers

Answered by ynagamalleswarao
3

Step-by-step explanation:

any positive integer is in the. form of 3q,3q+1,3q+2

if n=3q

n^2=(3q)^2

=9q^2

3(3q^2)

is in the form of 3m

m=3q^2

n=3q+1

n^2=(3q+1)^2

=3q^2+1^2+2(3q)(1)

9q^2+1+6q

9q^2+6q+1

3(3q^2+2q)+1

is in the form of 3m+1

m=3q^2+2q

n=3q+2

n^3=(3q+2)^2

3q^2+2^2+2(3q)(2)

9q^2+4+12q

3(3q^2+4q)+4

therefore it is in the form of 3q+4

therefore all positive in refers are in the form of 3q,3q+1 but not in any other form

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